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FNOL2 is a FORTRAN subroutine used for the numerical integration of a system of up to 30 ordinary differential equations. It uses double precision arithmetic in key locations. FNOL2 has the option of automatically varying the interval of integration, h, to hold the relative or absolute truncation error within bounds fixed by the user. A FORTRAN listing of the subroutine is included in this report.

We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We study the consistency of an alternative estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution. This smooth and match estimator (SME) bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this smooth and match estimation procedure leads to a $\sqrt{n}$-consistent estimator of the parameter of interest.

The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations reducible to algebraic equations is found. It depends on two arbitrary functions. The method for solving all these equations is given. The new class contains the constant coefficient equations and Euler's equations as particular cases.

A class of two-point boundary value problems are considered for systems for the form dx = f(x, y, t, epsilon), edy = g sub 1 (x, t, epsilon) + g sub 2 (x, t, epsilon)y where the matrix g sub 2 (x, t, o) has a hyperbolic splitting with a fixed number of stable and unstable eigenvalues. Solutions to such boundary value problems can then be expected to have boundary layer behavior near both endpoints in the limit epsilon approaches 0 to a + value. Our analysis shows, in particular, how to determine the reduced order boundary value problem satisfied by the limiting interior solution. Orthogonal matrix methods are used to determine this reduced problem and appropriate boundary layer corrections in a computationally effective manner. Numerical experiments with model problems illustrate the possibility of multiple solutions and show how our asymptotic results can be used in combination with the COLSYS code for solving two-point problems via collocation.

ASC DEQ4 is a floating point subroutine, written in FORTRAN IV source language, which integrates numerically a set of N simultaneous second-order ordinary differential equations in which first derivatives may or may not appear. If the N equations can be separated into two groups (IB and N-IB) such that the first IB equations are not dependent on the final N-IB equations (e.g., variational equations) then DEQ-4 has the capability of integrating the final N- IB equations at a larger step size than the first IB equations, thus saving 2(R- 1) (N-IB) derivatives per integration step. This subroutine obsoletes subroutine DEQ2 with the following improvements: better accuracy controls, new starting procedure, improved halving and doubling procedure, reduction in computing time, and reduction in core storage requirements (10N less). The subroutine is restricted in that it contains 20 digit octal constants (real constants) for the CDC 6000 series machines.

ASC DEQ4 is a floating point subroutine, written in FORTRAN IV source language, which integrates numerically a set of N simultaneous second-order ordinary differential equations in which first derivatives may or may not appear. If the N equations can be separated into two groups (IB and N-IB) such that the first IB equations are not dependent on the final N-IB equations (e.g., variational equations) then DEQ-4 has the capability of integrating the final N- IB equations at a larger step size than the first IB equations, thus saving 2(R- 1) (N-IB) derivatives per integration step. This subroutine obsoletes subroutine DEQ2 with the following improvements: better accuracy controls, new starting procedure, improved halving and doubling procedure, reduction in computing time, and reduction in core storage requirements (10N less). The subroutine is restricted in that it contains 20 digit octal constants (real constants) for the CDC 6000 series machines.

Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

http://uf.catalog.fcla.edu/uf.jsp?st=UF003875150&ix=nu&I=0&V=D&pm=1

We present a convergence proof of the projective integration method for a class of deterministic multi-dimensional multi-scale systems which are amenable to centre manifold theory. The error is shown to contain contributions associated with the numerical accuracy of the microsolver, the numerical accuracy of the macrosolver and the distance from the centre manifold caused by the combined effect of micro- and macrosolvers, respectively. We corroborate our results by numerical simulations.

We present a convergence proof of the projective integration method for a class of deterministic multi-dimensional multi-scale systems which are amenable to centre manifold theory. The error is shown to contain contributions associated with the numerical accuracy of the microsolver, the numerical accuracy of the macrosolver and the distance from the centre manifold caused by the combined effect of micro- and macrosolvers, respectively. We corroborate our results by numerical simulations.

31 p. 28 cm

Parameter estimation in ordinary differential equations, although applied and refined in various fields of the quantitative sciences, is still confronted with a variety of difficulties. One major challenge is finding the global optimum of a log-likelihood function that has several local optima, e.g. in oscillatory systems. In this publication, we introduce a formulation based on continuation of the log-likelihood function that allows to restate the parameter estimation problem as a boundary value problem. By construction, the ordinary differential equations are solved and the parameters are estimated both in one step. The formulation as a boundary value problem enables an optimal transfer of information given by the measurement time courses to the solution of the estimation problem, thus favoring convergence to the global optimum. This is demonstrated explicitly for the fully as well as the partially observed Lotka-Volterra system.

Operational unification of finite difference methods for numerical integration of ordinary differential equations

Self-starting multistep methods for numerical integration of ordinary differential equations

A study is made of the step-by-step integration of ordinary differential equations of the first order by means of formulas obtained from the Gregory-Newton backward interpolating formula. Tables of relevant constants are presented.

We present a convergence proof for higher order implementations of the projective integration method (PI) for a class of deterministic multi-scale systems in which fast variables quickly settle on a slow manifold. The error is shown to contain contributions associated with the length of the microsolver, the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro- and macrosolvers, respectively. We also provide stability conditions for the PI methods under which the fast variables will not diverge from the slow manifold. We corroborate our results by numerical simulations.

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that BODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution-of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method can be more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = ATN exp(-E/RT)) only when the temperature change exceeds an amount T that is problem dependent. An approximate expression for the automatic evaluation of T is derived and is shown to result in increased efficiency.

Error analyses for numerically integrating first order ordinary differential equations

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: 1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). 2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this subject. The techniques provided are important in applications for finding approximate solutions to problems for which analytic solutions either cannot be found or are too difficult to evaluate. (RP)

Operational unification of finite difference methods for numerical integration of ordinary differential equations

AN EXPOSITION OF PROCEDURES IS PRESENTED FOR APPROXIMATION OF PARTICULAR SOLUTIONS TO LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS (OR SYSTEMS OF EQUATIONS) BY MEANS OF DIFFERENCE EQUATIONS. THE RESULTS OF THE CLASSICAL AND MODIFIED PROCEDURES ARE COMPARED FOR SPECIFIC EXAMPLES. QUESTIONS OF CONVERGENCE, STABILITY (RELATIVE AND ABSOLUTE), CONSISTENCY, AND OPTIMALITY ARE TREATED.

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Conditions are given for a class of nonlinear ordinary differential equations x''(t)+a(t)w(x)=0, t>=1, which includes the linear equation to possess solutions x(t) with prescribed oblique asymptote that have an oscillatory pseudo-wronskian x'(t)-x(t)/t.

Examples and a criterion for stability of integration methods is provided. The criterion is applied to well-known integration formulas.

146p

New Lie series method for numerical integration of ordinary differential equations, and restricted three-body problem application

This note is concerned with the explicit symbolic computation of expressions involving differential operators and their actions on functions. The derivation of specialized numerical algorithms, the explicit symbolic computation of integrals of motion, and the explicit computation of normal forms for nonlinear systems all require such computations. More precisely, if R = k(x(sub 1),...,x(sub N)), where k = R or C, F denotes a differential operator with coefficients from R, and g member of R, we describe data structures and algorithms for efficiently computing g. The basic idea is to impose a multiplicative structure on the vector space with basis the set of finite rooted trees and whose nodes are labeled with the coefficients of the differential operators. Cancellations of two trees with r + 1 nodes translates into cancellation of O(N(exp r)) expressions involving the coefficient functions and their derivatives.

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

Examples and a criterion for stability of integration methods is provided. The criterion is applied to well-known integration formulas.

Examples and a criterion for stability of integration methods is provided. The criterion is applied to well-known integration formulas.

Source: Digital Library of India Scanning Centre: Allama Iqbal Library, University of Kashmir Source Library: Women College M.A Road Srinagar Date Accessioned: 7/14/2015 21:42 The Digital Library of India was a project under the auspices of the Government of India.

By revisiting an asymptotic integration theory of nonlinear ordinary differential equations due to J.K. Hale and N. Onuchic [Contributions Differential Equations 2 (1963), 61--75], we improve and generalize several recent results in the literature. As an application, we study the existence of bounded positive solutions to a large class of semi-linear elliptic partial differential equations via the subsolution-supersolution approach.